Factorials

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What is a Factorial?

In mathematics, the factorial of a non-negative integer n is the product of all positive integers less than or equal to n.

Mathematical Definition:

$$n!=n\times (n-1)\times (n-2) \times ...\times 3\times 2\times 1$$

Base Case:

$$0!=1$$

This is defined (not derived) so that formulas like permutations and combinations work correctly.

Examples:

n	n!	Explanation
0	1	Defined as 1
1	1	1
2	2	2 × 1
3	6	3 × 2 × 1
4	24	4 × 3 × 2 × 1
5	120	5 × 4 × 3 × 2 × 1

Step-by-Step: How Factorial Works

Let’s walk through how 4! is calculated step-by-step:

Step 1: Start with 4
Step 2: Multiply by 3 → 4 × 3 = 12
Step 3: Multiply by 2 → 12 × 2 = 24
Step 4: Multiply by 1 → 24 × 1 = 24

So:

4!=4×3×2×1=24

Factorial in Python (3 Ways)

1. Iterative Approach (Using a Loop)

def factorial_iterative(n):
    result = 1
    for i in range(1, n + 1):
        result *= i
    return result

print(factorial_iterative(5))  # Output: 120

Output

120

2. Recursive Approach (Mathematical Style)

def factorial_recursive(n):
    if n == 0 or n == 1:
        return 1
    return n * factorial_recursive(n - 1)

print(factorial_recursive(5))  # Output: 120

3. Using Python’s Built-in math.factorial()

import math

print(math.factorial(5))  # Output: 120

Visualizing Factorial Growth (with matplotlib)

import matplotlib.pyplot as plt
import math

x = list(range(10))  # 0! to 9!
y = [math.factorial(i) for i in x]

plt.plot(x, y, marker='o')
plt.title('Factorial Growth')
plt.xlabel('n')
plt.ylabel('n!')
plt.grid(True)
plt.show()

Output

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